Question

Найдите длины полуосей,координаты фокусов, эксцентриситет ∈ гиперболы . Изобразите гиперболу на чертеже. Пожалуйста помогите.

Answer 1

$\frac{x^2}{4} - \frac{y^2}{9} =1$
$\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} =1$
$a=2$
$b=3$
$k=0$
$h=0$
$(0;0)$
$\sqrt{13}$
$(2;0);(-2;0)$
$( \sqrt{13} ;0);(- \sqrt{13};0)$
$\frac{ \sqrt{13} }{2}$
$\frac{9 \sqrt{13} }{ \sqrt{13} }$
$y=$±$\frac{3}{2} x+0$
$y= \frac{3x}{2}$
$y=- \frac{3x}{2}$
$y= \frac{3x}{2} ;y=- \frac{3x}{2}$
Эти величины представляют важные значения для построения графика и анализа гиперболы.
Центр: $(0;0)$
Вершины: $(2;0);(-2;0)$
Фокусы: $( \sqrt{13} ;0);(- \sqrt{13} ;0)$
Эксцентриситет: $\frac{ \sqrt{13} }{2}$
Расстояние от фокуса до директрисы: $\frac{9 \sqrt{13} }{13}$
Асимптоты: $y= \frac{3x}{2} ;y=- \frac{3x}{2}$

Answer 2

Hyperbola Parameters and Graph

To find the lengths of the semi-axes, coordinates of the foci, and the eccentricity of the hyperbola, we can use the following information from the search results:

- The lengths of the semi-axes are denoted by a and b. - The coordinates of the foci are denoted by F1 and F2. - The eccentricity of the hyperbola is denoted by e.

Using the provided information, we can calculate these parameters and then proceed to draw the hyperbola.

Lengths of Semi-Axes, Coordinates of Foci, and Eccentricity

The lengths of the semi-axes, coordinates of the foci, and the eccentricity of the hyperbola are as follows:

- Length of semi-major axis (a): - From Source the length of the semi-major axis is denoted by a.

- Length of semi-minor axis (b): - From Source the length of the semi-minor axis is denoted by b.

- Coordinates of the foci (F1 and F2): - From Source the coordinates of the foci are denoted by F1 and F2.

- Eccentricity (e): - From Source the eccentricity of the hyperbola is denoted by e.

Hyperbola Graph

To graph the hyperbola, we can use the information about the parametric equations of the hyperbola from Source

x = sin(u) + 2v
y = cos(u) + 3v
z = 12v

By varying the parameters u and v within certain ranges, we can plot the points that satisfy these equations to visualize the hyperbola in 3D space.

I hope this helps! If you have any further questions or need additional assistance, feel free to ask.